Relationship between Chebyshev T and Gegenbauer C

Theorem

The following formula holds for $n \in \{1,2,3,\ldots\}$:
$$T_n(x)=\dfrac{n}{2} \displaystyle\lim_{\lambda \rightarrow 0} \dfrac{C_n^{\lambda}(x)}{\lambda},$$
where $T_n$ denotes a Chebyshev polynomial of the first kind and $C_n^{\lambda}$ denotes a Gegenbauer C polynomial.